Method for detecting line-to-line fault location in power network

ABSTRACT

The present invention relates to a method for detecting a line-to-line fault location in power network, and more particularly, detecting the line-to-line fault location by direct 3-phase circuit analysis without using a symmetrical component transformation, so even in an unbalanced 3-phase circuit, the line-to-line fault location can be accurately detected. In the method using direct 3-phase circuit analysis of this invention, inverse lemma is used to simplify matrix inversion calculations, thus the line-to-line fault location can be easily and accurately determined even in the case of an unbalanced network without symmetrical component transformation.

BACKGROUND OF THE INVENTION

(a) Field of the Invention

The present invention relates to a method for detecting a line-to-linefault location in power network, and more particularly, detecting theline-to-line fault location with direct 3-phase circuit analysis withoutusing a symmetrical component transformation, whereby even in anunbalanced 3-phase circuit the line-to-line fault location can beaccurately detected.

(b) Description of the Related Art

Rapid growth of economy has resulted in large scale of power systems,and an excessive increase in transmission and distribution networks ofelectric power systems causes many kinds of faults by various causes.Transmission and distribution networks of electric power systems areplaying very important roles as the links between the power suppliersand the consumers. However, because most of lines are exposed to air,lightning, contact of animals or mal-functioning of protection devicescauses many kinds of faults. When a line-to-line fault occurs, detectinga fault location rapidly and precisely separating the part of thenetwork including the fault location from the rest part of the networkuntil repairing the fault being finished is very important to minimizepower interruption rate and to provide highly reliable power supplyingservice and electric power of high quality.

Transformation of 3-phase networks to symmetrical component systems(symmetrical component transformation) is generally used in conventionalmethods for detecting the line-to-line fault locations. A 3-phasebalanced network can be transformed to a symmetrical component system,which has no coupling between sequences so that the systems of equationmay be solved easily. In other words, diagonal sequence impedancematrices are obtained in case of the balanced networks, thus sequencevoltage and current can be expressed without any coupling among thesequences.

The advantage of the above method is that it can be easily applied to abalanced network. Zero sequence, positive sequence and negative sequencecan be easily analyzed because they are not correlated, that is, thereis no couplings, or equivalently, mutual impedances among the sequences.

However, the above conventional method can only be applied to a balancednetwork, because the simplified equations of zero sequence, positivesequence and negative sequence, which are not correlated, can beobtained by symmetrical component transformation only in the balancednetwork. Thus, the conventional method is not available for unbalancedsystems.

SUMMARY OF THE INVENTION

The present invention has been made in an effort to solve the aboveproblems.

It is an object of the present invention to provide a method fordetecting a line-to-line fault location in a power network including afault resistance, not using the symmetrical component transformation butusing direct 3-phase circuit analysis. The line-to-line fault locationdetecting method of the present invention requires phase voltage andphase current of a fundamental frequency measured at a relay.

In the method using direct 3-phase circuit analysis of the presentinvention, matrix inverse lemma is applied to simplify matrix inversioncalculations, thus the line-to-line fault location can be easily andaccurately determined even in the case of an unbalanced network withoutusing the symmetrical component transformation.

By using the method of the present invention, the line-to-line faultlocation can be directly analyzed whether the 3-phase network isbalanced or unbalanced.

To achieve the above object, the present invention provides a method fordetecting a line-to-line fault location in a power network comprisingthe steps of:

determining elements of a line impedance matrix and a load impedancematrix, and phase voltages and currents at a relay;

determining a line-to-line fault distance d by substituting saidelements of said line impedance matrix and said load impedance matrix,and said phase voltage and current into a fault location equation basedon direct circuit analysis;

wherein said fault location equation is derived from a model consistingof said phase voltage and current at the relay, a fault current, a faultresistance and the line-to-line fault distance;

wherein the model is based on the line-to-line fault between a-phase andb-phase and described by a model equation:V _(Sa) −V _(Sb)=(1−d)((Zl _(aa) −Zl _(ba))I _(Sa)+(Zl _(ab) −Zl _(bb))I_(Sb +() Zl _(ac) −Zl _(cb))I _(Sc))+l _(f) R _(f),

where, V_(Sabc)=[V_(sa) V_(Sb) V_(Sc)]′: phase voltage vector,I_(Sabc)=[I_(Sa) I_(Sb) I_(sc) ]′: phase current vector,${Zl}_{abc} = {\begin{bmatrix}{Zl}_{aa} & {Zl}_{ab} & {Zl}_{ac} \\{Zl}_{ba} & {Zl}_{bb} & {Zl}_{bc} \\{Zl}_{ca} & {Zl}_{cb} & {Zl}_{cc}\end{bmatrix}\text{:}}$line impedance matrix, I_(f): fault current, 1−d: fault distance;

wherein said fault location equation is derived by using the matrixinverse lemma: (A⁻¹+BCD)⁻¹=A−AB(C⁻¹+DAB)⁻¹DA, to simplify an inversematrix calculation; and

wherein the fault location equation is derived by direct circuitanalysis without using the conventional symmetrical componenttransformation method.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate an embodiment of the invention,and, together with the description, serve to explain the principles ofthe invention.

FIG. 1 shows a simple diagram of a line-to-line fault occurred in ageneral 3-phase power network whether it is balanced or unbalanced one;

FIG. 2 shows a flow of a preferred embodiment of the present invention;

FIG. 3 shows an example of the unbalanced system where a line-to-linefault between a-phase and b-phase occurs;

FIG. 4 shows errors resulted by using conventional method in case ofline-to-line fault for the unbalanced system; and

FIG. 5 shows errors resulted by using the proposed method in the samecase as in FIG. 4.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Preferred embodiment of the present invention will now be described indetail, with reference to the accompanying drawings.

FIG. 1 shows general 3-phase transmission and distribution of simplifieddiagram of faulted network whether they are operated in a balance orunbalance manner. The proposed algorithm use voltages and currentsmeasured at a relay.

Based on a model shown in FIG. 1 and assuming the capacitance of linesnegligible, phase voltages and phase currents at the location of relay Asatisfy the following model equation.V _(Sa) −V _(Sb)=(1−d)((Zl _(aa) −Zl _(ba))I _(sb)+(Zl _(ac) −Zl _(bb))I_(sb) +(Zl _(ac) −Zl _(cb))+I _(f) R _(f)  (1)

Where, V_(Sa) and V_(Sb) are phase voltages of a-phase and b-phaserespectively, and I_(Sa), I_(Sb) and I_(Sc) are phase currents ofa-phase, b-phase and c-phase respectively at the relay A,I_(Sabc)=[I_(Sa) I_(Sb) I_(Sc)]′ is a phase current vector at the relayA, ${Zl}_{abc} = \begin{bmatrix}{Zl}_{aa} & {Zl}_{ab} & {Zl}_{ac} \\{Zl}_{ba} & {Zl}_{bb} & {Zl}_{bc} \\{Zl}_{ca} & {Zl}_{cb} & {Zl}_{cc}\end{bmatrix}$represents a line impedance matrix, I_(f) represents a fault current,R_(f) is a fault resistance, d is a fault distance.

In equation (1), the line impedance is given, and the phase voltages andcurrents at relay can be measured. However, the fault current I_(f) andfault resistance R_(f) are unknown. They can be obtained using thedirect 3-phase circuit analysis of this invention as described below.

The fault current I_(f) can be represented as a function of the phasecurrents at the relay using current distribution law of a paralleladmittance network: $\begin{matrix}{\begin{bmatrix}I_{f} \\0 \\0\end{bmatrix} = {{Y_{f}\left\lbrack {Y_{f} + \left( {{dZl}_{abc} + {Zr}_{abc}} \right)^{- 1}} \right\rbrack}^{- 1}\begin{bmatrix}I_{Sa} \\I_{Sb} \\I_{Sc}\end{bmatrix}}} & (2)\end{matrix}$where, $Y_{f} = {\begin{bmatrix}{1/R_{f}} & {{- 1}/R_{f}} & 0 \\{{- 1}/R_{f}} & {1/R_{f}} & 0 \\0 & 0 & 0\end{bmatrix}\text{:}}$the fault admittance matrix at the fault location; and${Zr}_{abc} = {\begin{bmatrix}{Zr}_{aa} & {Zr}_{ab} & {Zr}_{ac} \\{Zr}_{ba} & {Zr}_{bb} & {Zr}_{bc} \\{Zr}_{ca} & {Zr}_{cb} & {Zr}_{cc}\end{bmatrix}\text{:}}$the load impedance matrix.

The inverse matrix [Y_(f)+(dZl_(abc)+Zr_(abc))⁻¹]⁻¹ in Eq. (2) can besimplified by the matrix inverse lemma:(A ⁻¹ +BCD)⁻¹ =A−AB(C ⁻¹ +DAB)⁻¹ DA  (3)

A, B, C and D are defined as follows: $\begin{matrix}{{A \equiv \left( {{dZl} + {Zr}} \right)} = \begin{bmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{bmatrix}} & (4) \\\begin{matrix}{{B \equiv \begin{bmatrix}1 \\{- 1} \\0\end{bmatrix}},} \\{{C \equiv {1/R_{f}}},} \\{D \equiv \left\lbrack {1\mspace{14mu} - {1\mspace{31mu} 0}} \right\rbrack}\end{matrix} & (5)\end{matrix}$

Then, applying the lemma, we obtain: $\begin{matrix}{\left. \left\lbrack {Y_{f} + {\left( {{dZl} + {Zr}} \right)^{- 1}{Zr}}} \right)^{- 1} \right\rbrack^{- 1} = {{\left\lbrack \begin{matrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{matrix} \right\rbrack - {\left\lbrack \begin{matrix}{a_{11} - a_{12}} \\{a_{21} - a_{22}} \\{a_{31} - a_{32}}\end{matrix} \right\rbrack{\left( {R_{f} + a_{11} + a_{22} - a_{12} - a_{21}} \right)^{- 1}\left\lbrack {a_{11} - {a_{21}\mspace{11mu} a_{12}} - {a_{22}\mspace{14mu} a_{13}} - a_{23}} \right\rbrack}}} = {{\left\lbrack \begin{matrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{matrix} \right\rbrack - {\frac{1}{\left( {R_{f} + a_{11} + a_{22} - a_{12} - a_{21}} \right)} \times \left\lbrack \begin{matrix}{{\left( {a_{11} - a_{12}} \right)\left( {a_{11} - a_{21}} \right)},} & {{\left( {a_{11} - a_{12}} \right)\left( {a_{12} - a_{22}} \right)},} & {\left( {a_{11} - a_{12}} \right)\left( {a_{13} - a_{23}} \right)} \\{{\left( {a_{21} - a_{22}} \right)\left( {a_{11} - a_{21}} \right)},} & {{\left( {a_{21} - a_{22}} \right)\left( {a_{12} - a_{22}} \right)},} & {\left( {a_{21} - a_{22}} \right)\left( {a_{13} - a_{23}} \right)} \\{xx} & {xx} & {xx}\end{matrix} \right\rbrack}} = {\frac{1}{\left( {R_{f} + a_{11} + a_{22} - a_{12} - a_{21}} \right)} \times \left\lbrack \begin{matrix}{\left( {{R_{f}a_{11}} + {a_{11}a_{22}} - {a_{12}a_{21}}} \right),} & {\left( {{R_{f}a_{12}} + {a_{11}a_{22}} - {a_{21}a_{12}}} \right),} & \left( {{R_{f}a_{13}} + {a_{22}a_{13}} - {a_{21}a_{13}} + {a_{23}a_{11}} - {a_{23}a_{12}}} \right) \\{\left( {{R_{f}a_{21}} + {a_{11}a_{22}} - {a_{12}a_{21}}} \right),} & {\left( {R_{f} + {a_{11}a_{22}} - {a_{12}a_{21}}} \right),} & \left( {{R_{f}a_{23}} + {a_{11}a_{23}} - {a_{12}a_{23}} - {a_{13}a_{21}} + {a_{13}a_{22}}} \right) \\{xx} & {xx} & {xx}\end{matrix} \right\rbrack}}}} & (6)\end{matrix}$

Then the fault current equation, Eq. (2) can be rewritten as:$\begin{matrix}{\begin{bmatrix}I_{f} \\0 \\0\end{bmatrix} = {{\frac{1}{\left( {R_{f} + C_{1} - C_{2}} \right)}\begin{bmatrix}C_{1} & C_{2} & C_{3} \\{xx} & {xx} & {xx} \\{xx} & {xx} & {xx}\end{bmatrix}}\begin{bmatrix}I_{Sa} \\I_{Sb} \\I_{Sc}\end{bmatrix}}} & (7)\end{matrix}$

where, xx represents an element that we have no concern and othercoefficients are as follows:C ₁ =a ₁₁ −a ₂₁ =d(Zl _(aa) −Zl _(ba))+Zr _(aa) −Zr _(ba) =dA ₁ +B ₁ ,C ₂ =a ₁₂ −a ₂₂ =d(Zl _(ab) −Zl _(bb))+Zr _(ab) −Zr _(bb) =dA ₂ +B ₂ andC ₃ =a ₁₃ −a ₂₃ =d(Zl _(ac) −Zl _(bc))+Zr _(ac) −Zr _(bc) =dA ₃ +B ₃.  (8)

Thus, the final expression for the fault current becomes:$\begin{matrix}{I_{f} = \frac{{C_{1}I_{Sa}} + {C_{2}I_{Sb}} + {C_{3}I_{Sc}}}{\left( {R_{f} + C_{1} - C_{2}} \right)}} & (9)\end{matrix}$

Substitution of Eq. (9) into the model equation of Eq. (1) andrearrangement can make a second order polynomial with respect to thedistance variable d using expressions of the coefficients as in Eq. (8).d ²(a _(r) +ja _(i))+d(b _(r) +jb _(i))+c _(r) +jc _(i) +R _(f)(d _(r)+jd _(i))=0  (10)where,a _(r) +ja _(i)=(A ₁ −A ₂)D ₁,b _(r) +jb _(i)=(A ₁ −A ₂)(V _(Sa) −V _(Sb) −D ₁)+(B ₁ −B ₂)D ₁,c _(r) +jc _(i)=(B ₁ −B ₂)(V _(Sa) −V _(Sb) −D ₁),d _(r) +jd _(i)=(V _(Sa) −V _(Sb) −D ₁ −D ₂),D ₁ =A ₁ I _(Sa) +A ₂ I _(Sb) +A ₃ I _(Sc) andD ₂ =B ₁ I _(Sa) +B ₂ I _(Sb) +B ₃ I _(Sc).  (11)

From the imaginary part of Eq. (10), the following second orderpolynomial equation can be obtained. $\begin{matrix}{{{d^{2}\left( {a_{r} - {\frac{\mathbb{d}_{r}}{\mathbb{d}_{i}}a_{i}}} \right)} + {d\left( {b_{r} - {\frac{\mathbb{d}_{r}}{\mathbb{d}_{i}}b_{i}}} \right)} + c_{r} - {\frac{\mathbb{d}_{r}}{\mathbb{d}_{i}}c_{i}}} = 0} & (12)\end{matrix}$

Finally the fault distance d can be obtained by solving Eq. (12). Notethat this fault location equation based on the direct circuit analysiscan be applied to any system, which is balanced or unbalanced type,three phase or three/single phase systems.

FIG. 2 shows a flow chart of the preferred embodiment of this invention.

The preferred embodiment shown in FIG. 2 comprises the steps of:determining elements of a line impedance matrix and a load impedancematrix, and phase voltages and currents at a relay (step S10);determining a line-to-line fault distance d by substituting saidelements of said line impedance matrix and said load impedance matrix,and said phase voltages and currents into a fault location equationbased on direct circuit analysis (step S20); and outputting theline-to-line fault distance d to the network protecting device such asprotective relays (step S30).

The suggested method based on direct circuit analysis for theline-to-line fault has been also applied to the unbalanced system inFIG. 3 for verification. The line-to-line fault is assumed to haveoccurred between A and B. The results are compared with those of theconventional method using the distribution factor. FIG. 4 shows errorsresulted by using the conventional method in case of line-to-line faultfor the unbalanced system, while the FIG. 5 shows errors resulted byusing the proposed method in the same case as in FIG. 4.

A significant accuracy difference can be observed between two results.The maximum estimation error is 8% in case of the conventional methodwhile 0.15% in case of the proposed method. The error in the proposedmethod is very small showing its effectiveness for the real application.

A new fault location algorithms based on the direct circuit analysis aresuggested. Application of the matrix inverse lemma has greatlysimplified the derivation of the fault location equations that,otherwise, are too complicated to be derived. The proposed algorithmsovercome the limit of the conventional fault location algorithms basedon the sequence circuit analysis, which assumes the balanced systemrequirement. The proposed algorithms are applicable to any power system,but especially useful for the unbalanced distribution systems. Itseffectiveness has been proved through many EMTP simulations. Theobjective of the embodiments and drawings is to clearly explain thepresent invention and does not limit the technical scope of theinvention. The present invention described above can be replaced,modified and changed by one skilled in the art, as long as such changesdo not exceed the technical scope of the invention. It is, therefore, tobe understood that the appended claims are intended to cover all suchmodifications and changes as fall within the true spirit of theinvention.

1. A method for detecting a line-to-line fault location in a powernetwork comprising the steps of: determining elements of a lineimpedance matrix and a load impedance matrix, and phase voltages andcurrents at a relay; determining a line-to-line fault distance d bysubstituting said elements of said line impedance matrix and said loadimpedance matrix, and said phase voltage and current into a faultlocation equation based on direct circuit analysis; wherein said faultlocation equation is derived from a model consisting of said phasevoltage and current at the relay, a fault current, a fault resistanceand the line-to-line fault distance; wherein the model is based on theline-to-line fault between a-phase and b-phase and described by a modelequation:V _(Sa) −V _(Sb)=(1−d)((Zl _(aa) −Zl _(ba))I _(Sa)+(Zl _(ab) −Zl _(bb))I_(Sb) +(Zl _(ac) −Zl _(cb))+I _(f) R _(f), where, V_(Sabc)=[V_(Sa)V_(Sb) V_(Sc)]′: phase voltage vector, I_(Sabc)=[I_(Sa) I_(Sb) I_(Sc)]′: phase current vector, ${Zl}_{abc} = {\begin{bmatrix}{Zl}_{aa} & {Zl}_{ab} & {Zl}_{ac} \\{Zl}_{ba} & {Zl}_{bb} & {Zl}_{bc} \\{Zl}_{ca} & {Zl}_{cb} & {Zl}_{cc}\end{bmatrix}:}$ line impedance matrix, I_(f): fault current, 1−d: faultdistance; wherein said fault location equation is derived by using thematrix inverse lemma: (A⁻¹+BCD)⁻¹=A−AB(C⁻¹+DAB)⁻¹DA, to simplify aninverse matrix calculation; and wherein the fault location equation isderived by direct circuit analysis without using the conventionalsymmetrical component transformation method.
 2. The method of claim 1,wherein the power network is a 3-phase balanced network.
 3. The methodof claim 1, wherein the power network is a 3-phase unbalanced network.4. The method of claim 1, wherein the fault location equation is derivedby steps of: (a) expressing the fault current If in terms of the phasecurrent vector I_(s) by using current distribution law of a parallelnetwork yielding: ${\begin{bmatrix}I_{f} \\0 \\0\end{bmatrix} = {{Y_{f}\left\lbrack {Y_{f} + \left( {{dZl}_{abc} + {Zr}_{abc}} \right)^{- 1}} \right\rbrack}^{- 1}\begin{bmatrix}I_{Sa} \\I_{Sb} \\I_{Sc}\end{bmatrix}}},{{{where}\mspace{20mu} Y_{f}} = {\begin{bmatrix}{1/R_{f}} & {{- 1}/R_{f}} & 0 \\{{- 1}/R_{f}} & {1/R_{f}} & 0 \\0 & 0 & 0\end{bmatrix}:}}$ fault admittance matrix and${Zr}_{abc} = {\begin{bmatrix}{Zr}_{aa} & {Zr}_{ab} & {Zr}_{ac} \\{Zr}_{ba} & {Zr}_{bb} & {Zr}_{bc} \\{Zr}_{ca} & {Zr}_{cb} & {Zr}_{cc}\end{bmatrix}:}$ load impedance matrix; (b) simplifying the equation ofstep (a) by using the inverse matrix lemma and substituting thesimplified equation into the model equation; (c) deriving a second orderpolynomial equation with respect to the line-to-line fault distance dfrom a real or an imaginary part of the equation obtained at step (b);and (d) deriving the fault location equation by solving the second orderpolynomial equation.